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Square root functions are a special type of function that involve square roots. A square root function consists of expressions like \( \sqrt{a} \), where \( a \) must be greater than or equal to zero. This is because the square root of a negative number is not defined within the realm of real numbers. In the function given in the exercise, \( \sqrt{x+4} \) and \( \sqrt{1-x} \) are two such expressions.
To find the domain of any function with square roots, begin by ensuring that the expressions under the square roots are non-negative. For example:

• For \( \sqrt{x+4} \), evaluate \( x + 4 \geq 0 \) which simplifies to \( x \geq -4 \).
• For \( \sqrt{1-x} \), evaluate \( 1-x \geq 0 \) which simplifies to \( x \leq 1 \).

In summary, both of these conditions must be fulfilled: \( x \geq -4 \) and \( x \leq 1 \) so that the square root functions are defined and valid for the inputs.

Interval notation is a concise method to describe a set of numbers, specifically those that fall within a certain range. It is commonly used to define the domain of functions. Consider the solution's constraints: \( x \geq -4 \) and \( x \leq 1 \) while excluding \( x = 0 \). Interval notation helps describe this domain in a simple format.
When describing intervals:

• Use brackets \( [ ] \) to indicate the inclusion of endpoints.
• Use parentheses \( ( ) \) to indicate the exclusion of endpoints.

For the function in our exercise, the domain between \( x \geq -4 \) and \( x \leq 1 \) can be expressed as \([-4, 1]\). However, since \( x eq 0 \), it excludes zero, so it is written as two separate intervals: \([-4, 0) \cup (0, 1]\). This simply means the set includes all x-values from -4 up to but not including 0, and from just above 0 to including 1.

Division by zero is a rule in mathematics that must never be violated. It's an undefined operation because there's no real number that you can multiply by zero to yield a non-zero result. In the context of the function \( f(x)=\sqrt{x+4}-\frac{\sqrt{1-x}}{x} \), division by zero occurs in the term \( \frac{\sqrt{1-x}}{x} \) if \( x = 0 \).
This particular condition requires attention because division by zero would render the function undefined at that point. As a rule of thumb when dealing with any rational expressions involving division, ensure that the denominator is never zero. For this exercise, \( x eq 0 \) is a key constraint we must satisfy. It's essential to remember:

• Identify wherever a variable appears in a denominator and determine the values for which the denominator is zero.
• Exclude these values from the domain of the function.

Thus, for \( \frac{\sqrt{1-x}}{x} \), ensure \( x eq 0 \), and this is reflected in the interval notation as the exclusion in \([-4, 0) \cup (0, 1]\).